1.5: Dividing Fractions
- Page ID
- 7085
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)When dividing fractions you invert (flip upside down) the fraction on the right side of the equation (the dividend). Then it becomes a multiplication problem. Invert and multiply!
Example \(\PageIndex{1}\)
\(\dfrac{2}{5} \div \dfrac{1}{2} \rightarrow \dfrac{2}{5} \times \dfrac{2}{1}=\dfrac{4}{5}\)
Example \(\PageIndex{2}\)
\(\dfrac {\dfrac{4}{9}}{\dfrac{8}{9}}\)
This example reads \(\dfrac{4}{9}\) divided by \(\dfrac{8}{9}\).
However, after you “invert and multiply,” it becomes:
\[\dfrac{4}{9} \times \dfrac{9}{8} \rightarrow \dfrac{1}{1} \times \dfrac{1}{2}=\dfrac{1}{2} \nonumber \]
After inverting the fraction, the same rules apply as previously mentioned when multiplying fractions. You need to change mixed numbers into improper fractions; you can cross cancel, and always remember to reduce if necessary.
Example \(\PageIndex{3}\)
\(\dfrac{3}{8} \div \dfrac{12}{6} \rightarrow \dfrac{3}{8} \times \dfrac{6}{12} \rightarrow \dfrac{\not{3}}{\not {8}} \times \dfrac{\not {6}}{\not {12}}=\dfrac{1}{4} \times \dfrac{3}{4}=\dfrac{3}{16}\)
In the Example \(\PageIndex{3}\) above the 3 divided into itself once and into the 12 four times. Similarly, 2 divided into 6, three times and into 8, four times. Can you see a different way to cross cancel?
If the dividend is a whole number write it as a fraction before inverting. Always remember to cross cancel and reduce if necessary.
Example \(\PageIndex{4}\)
\(\dfrac{5}{8} \div 10 \rightarrow \dfrac{5}{8} \div \dfrac{10}{1} \rightarrow \dfrac{5}{8} \times \dfrac{1}{10} \rightarrow \dfrac{\not{5}}{8} \times \dfrac{1}{\not {10}} \rightarrow \dfrac{1}{8} \times \dfrac{1}{2}=\dfrac{1}{16}\)
Exercise 1.5
Divide the following and reduce if necessary.
- \(\dfrac{1}{2} \div \dfrac{2}{4}\)
- \(\dfrac{5}{22} \div 7 \dfrac{2}{4}\)
- \(\dfrac{6}{8} \div \dfrac{9}{12}\)
- \(3 \dfrac{1}{3} \div 10\)
- \(5 \dfrac{1}{4} \div 8 \dfrac{1}{2}\)
- \(\dfrac{8}{16} \div \dfrac{16}{8}\)
- \(4 \div 3 \dfrac{2}{3}\)
- \(\dfrac{2}{5} \div 5 \dfrac{6}{9}\)
- \(\dfrac{9}{13} \div 9\)
- \(\dfrac{11}{22} \div \dfrac{1}{2}\)
- \(2 \dfrac{9}{20} \div 5 \dfrac{2}{5}\)
- \(3 \dfrac{1}{2} \div 9 \dfrac{1}{2}\)
- \(5 \div 25\)
- \(\dfrac{10}{3} \div \dfrac{1}{6}\)
- \(\dfrac{13}{33} \div \dfrac{39}{3}\)
- \(100 \dfrac{1}{2} \div 10 \dfrac{5}{6}\)